Question

A steel rod of 12 mm in diameter and 60 mm long with an insulated end is to be used as a spine. It is exposed to surroundings with a temperature of ${60}^{\circ }C$ and a heat transfer coefficient of $55W/m^2{}^{\circ }C.$ For this fin the efficiency is found that $62.18\%$, then the effectiveness of the fin is

Answer 1

$\begin{array}{cc}{\epsilon }_{fin}& =\frac{Q_{withfin}}{Q_{withoutfin}}\\ {\eta }_{fin}& =\frac{Q_{actual}}{Q_{max}}\\ \Rightarrow Q_{actual}& ={\eta }_{fin}\times Q_{max}\\ {\epsilon }_{fin}& =\frac{{\eta }_{fin}\times Q_{max}}{Q_{withoutfin}}\\ \text{For insulated tip,}& \\ Q_{max}& =h\left(PL\right){\theta }_o\\ Q_{withoutfin}& =hA{\theta }_o\\ {\epsilon }_{fin}& =\frac{{\eta }_{fin}\times h\left(PL\right){\theta }_0}{hA{\theta }_o}\\ & =\frac{{\eta }_{fin}\times \pi DL}{\frac{\pi }{4}{D}^2}=\frac{\eta \left(4L\right)}{D}\\ & =\frac{0.6218\times 4\times 0.06}{0.012}=12.436\end{array}$